$$\int\limits_0^2 {\left[ {{x^2}} \right]dx} $$   is-

Solution :
$$\eqalign{ & \int\limits_0^2 {\left[ {{x^2}} \right]dx} = \int\limits_0^1 {\left[ {{x^2}} \right]dx} + \int\limits_1^{\sqrt 2 } {\left[ {{x^2}} \right]dx} + \int\limits_{\sqrt 2 }^{\sqrt 3 } {\left[ {{x^2}} \right] + } \int\limits_{\sqrt 3 }^2 {\left[ {{x^2}} \right]dx} \cr & = \int\limits_0^1 {0\,dx + } \int\limits_1^{\sqrt 2 } {1\,dx + } \int\limits_{\sqrt 2 }^{\sqrt 3 } {2\,dx} + \int\limits_{\sqrt 3 }^2 {3\,dx} \cr & = \left[ x \right]_1^{\sqrt 2 } + \left[ {2x} \right]_{\sqrt 2 }^{\sqrt 3 } + \left[ {3x} \right]_{\sqrt 3 }^2 \cr & = \sqrt 2 - 1 + 2\sqrt 3 - 2\sqrt 2 + 6 - 3\sqrt 3 \cr & = 5 - \sqrt 3 - \sqrt 2 \cr} $$